I'm having a hard time figuring out which is the correct form of asking this kind of question. I mean speaking strictly, this doesn't sound right: You alright? or You eaten anything? compared to Are you alright? and Have you eaten anything?.
So please enlighten me. Are those both forms correct or just something which is ignored for the sake slang speakers?

These aren't correct in formal English. Here, the sentences say "Are you alright?", "Have you eaten anything", as you mentioned. The "Are"/"Have" are implied here, that's all. One finds this a lot in informal English -- words which can be guessed from context are sometimes dropped altogether.

It's just a way to save time, similar to how contractions are used (though contractions may be used in formal English as well)

“You alright?” is not necessarily true according to the grammar rules. It is instead used for conversations with friends and relatives. The most correct sentence, in my opinion, would definitely be: “Are you alright?”